Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advection–diffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection–diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates.     

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Matching and multiscale expansions for a model singular perturbation problem

We consider the Laplace–Dirichlet equation in a polygonal domain which is perturbed at the scale ε near one of its vertices. We assume that this perturbation is self-similar, that is, derives from the same pattern for all values of ε. On the base of this model problem, we compare two different approaches: the method of matched asymptotic expansions and the method of multiscale expansion. We enlighten the specificities of both techniques, and show how to switch from one expansion to the other. To cite this article: S. Tordeux et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Stress continuity in DEM-FEM multiscale coupling based on the generalized bridging domain method

Concurrent multiscale method is a spatial and temporal combination of two different scale models for describing the micro/meso and macro mixed behaviors observed in strain localization, failure and phase transformation processes, etc. Most of the existing coupling schemes use the displacement compatibility conditions to glue different scale models, which leads to displacement continuity and stress discontinuity for the obtained multiscale model. To overcome stress discontinuity, this paper presented a multiscale method based on the generalized bridging domain method for coupling the discrete element (DE) and finite element (FE) models. This coupling scheme adopted displacement and stress mixed compatibility conditions. Displacements that were interpolated from FE nodes were prescribed on the artificial boundary of DE model, while stresses at numerical integration points that were extracted from DE contact forces were applied on the material transition zone of FE model (the coupling domain and the artificial boundary of FE model). In addition, this paper proposed an explicit multiple time-steps integration algorithm and adopted Cundall nonviscous damping for quasi-static problems. DE and FE parameters were calibrated by DE simulations of a biaxial compression test and a deposition process. Numerical examples for a 2D cone penetration test (CPT) show that the proposed multiscale method captures both mesoscopic and macroscopic behaviors such as sand soil particle rearrangement, stress concentration near the cone tip, shear dilation, penetration resistance vibration and particle rotation, etc, during the cone penetration process. The proposed multiscale method is versatile for maintaining stress continuity in coupling different scale models.     

Improved interface conditions for a non-overlapping domain decomposition of a non-convex polygonal domain

We propose a local improvement of domain decomposition methods which fits with the singularities occurring in the solutions of elliptic equations in polygonal domains. This short presentation focuses on a model elliptic problem with the decomposition of a non-convex polygonal domain into convex polygonal subdomains. After explaining the strategy and the theoretical design of adapted interface conditions at the corner, we present numerical experiments which show that these new interface conditions satisfy some optimality properties. To cite this article: C. Chniti et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Une nouvelle méthode de raffinement de maillage spatio-temporel pour l'équation des ondes

In the present Note we introduce an extension of the conservative space–time mesh refinement method presented by Fouquet et al. We also propose a post-treatment of the solution that reduces the spurious phenomena due to the non-conformity between the time meshes. A reinterpretation of the equations in terms of new unknowns leads to a new scheme with second order consistent coupling equations. Numerical experiments in 2D and a plane wave analysis for the 1D model show that the method is second order accurate for an arbitrary refinement. To cite this article: J. Rodríguez, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Multiscale support vector regression method in Sobolev spaces on bounded domains

In this paper, we investigate the multiscale support vector regression (SVR) method for approximation of functions in Sobolev spaces on bounded domains. The Vapnik ?-intensive loss function, which has been developed well in learning theory, is introduced to replace the standard l² loss function in multiscale least squares methods. Convergence analysis is presented to verify the validity of the multiscale SVR method with scaled versions of compactly supported radial basis functions. Error estimates on noisy observation data are also derived to show the robustness of our proposed algorithm. Numerical simulations support the theoretical predictions.     

Multiscale Coupling through Locally Enriched Finite Elements

Multiscale modeling helps us to focus our limited computational power into those special places where traditional models based on continuum mechanics will fail while not losing the big picture of the macro scale behavior. An hourglass shaped development can be observed in today's simulation technologies. Simulation tools in the macroscale category and that for the micro phenomenons are both relatively well developed. Many algorithms and methods have been proposed in recent years to fill the gap between them. However, rather than trying to bridging different techniques, many tend to replace them completely and become independent simulation tools. Since many single–scale models have already been widely adopted by both the industry and the academy, it would be more beneficial to concentrate just on coupling techniques which can be applied without significant modifications of the original simulation framework. In this work, we present a multiscale idea of coupling the fine–scale model with the coarse–scale model through local enrichment within the elements at the coupling boundary. Higher order shape functions have been used to ‘enrich’ the coarse–scale model, allowing softer transition of the displacement field from the fine–scale model to the coarse–scale model. A least–square process has been used to fit the displacement gradients of different models at the coupling region. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving Diffusion Problems on Rough Surfaces with a Hierarchical Multiscale FEM

Diffusion on rough surfaces is a basic problem for many applications in engineering and the sciences. Solving these problems with a standard finite element method is often difficult or even impossible, due to the computational work and the amount of memory needed to triangulate the whole surface with a mesh which resolves its oscillations. We discuss in this paper a hierarchical Finite Element Method of “heterogeneous multiscale” type, which only needs to resolve the surface's fine scale on small sampling domains within a macro triangulation of the underlying smooth surface. This method converges, for periodic surface roughness and sufficiently small amplitude, at a robust (i.e. scale independent) rate, to the homogenized solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Évaluation ponctuelle et espace de Hardy : le cas multi-échelle

We define a point evaluation for transfer operators of multiscale causal dissipative systems. We associate to such a system a de Branges Rovnyak space, which serves as the state space of a coisometric realization. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A Large-Deformation Phase-Field Approach for the Modeling of Magneto-Sensitive Elastomers

Magneto-sensitive materials show magneto-mechanical coupled response and are thus of increasing interest in the recent age of smart functional materials. Ferromagnetic particles suspended in an elastomeric matrix show realignment under the influence of an external applied field, in turn causing large deformations of the substrate material. The magneto-mechanical coupling in this case is governed by the magnetic properties of the inclusion and the mechancial properties of the matrix. The magnetic phenomenon in ferromagnetic materials is governed by the formation and evolution of domains on the micro scale. A better understanding of the behavior of these particles under the influence of an external applied field is required to accurately predict the behavior of such materials. In this context it is of particular importance to model the macro scopic magneto-mechanically coupled behavior based on the micro-magnetic domain evolution. The key aspect of this work is to develop a large-deformation micro-magnetic model that can accurately capture the microscopic response of such materials. Rigorous exploitation of appropriate rate-type variational principles and consequent incremental variational principles directly give us field equations including the time evolution equation of the magnetization, which acts as the order parameter in our formulation. The theory presented here is the continuation of the work presented in [1, 7] for small deformations. A summary of magneto-mechanical theories spanning over multiple scales has been presented in [4]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation du drapé des tissus par maillages adaptatifsAdaptive meshing for cloth draping

Numerical simulation methods regarding fabric and cloth draping are generally based on mechanical models. These models are usually applied to uniform grids representing the true geometry of the fabric. Fabrics being a very flexible material, wrinkles appear on its surface when submitted to free or constrained motion. The main problem of the simulation is to represent realistically cloth surface motion. This is strongly dependent on the surface discretization. We present a new cloth animation scheme based on adaptive surface discretization. It can be seen as a multi-grid method which allows us to obtain realistic simulations. We propose also a new mechanical model well suited to our adaptive meshing strategy. A numerical example is given to show the efficiency of the method. To cite this article: J. Villard et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 561–566.     

Holonomic systems with solutions ramified along a cuspSystèmes holonomes avec solutions ramifiées le long d'un cusp

We classify the holonomic systems of (micro) differential equations of multiplicity one along a singular Lagrangian irreducible variety contained in an involutive submanifold of maximal codimension. We show that their solutions are related to _kF_k?1 hypergeometric functions on the Riemann sphere. To cite this article: O. Neto, P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 171–176.     

Multiscale model for moisture transport with adsorption phenomenon in concrete materials

Abstract

We consider a new two-scale problem which is given as a mathematical model for moisture transport arising in a concrete carbonation process. In research for moisture transport, it is a crucial step how to describe the relationship between the relative humidity and the degree of saturation, mathematically. Here, we have proposed the two-scale model consisting of the diffusion equation for the relative humidity in a macro domain and the free boundary problems describing the relationship in infinitely micro domains. Accordingly, the structures of the micro domains are unknown in our model. This is a significant feature of our new model to emphasize. The aim of this paper is to establish local existence in time and uniqueness of a solution to the model.     

Processus stationnaires sur l'arbre dyadique : prédiction et extension de covariance

We define for multiscale dyadic stationary processes the notion of one step positive extension of the covariance matrix, which is the counterpart of the central extension in the single scale case. To cite this article: D. Alpay, D. Volok, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Denoising using nonlinear multiscale representations

The goal of this paper is to present some numerical results for the one-dimensional denoising problem by using the nonlinear multiscale representations. We introduce modified thresholding strategies in this new context which give significant significant improvements for one-dimensional denoising problems. To cite this article: B. Matei, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A proximal approach to the inversion of ill-conditioned matrices

We propose a general proximal algorithm for the inversion of ill-conditioned matrices. This algorithm is based on a variational characterization of pseudo-inverses. We show that a particular instance of it (with constant regularization parameter) belongs to the class of fixed point methods. Convergence of the algorithm is also discussed. To cite this article: P. Maréchal, A. Rondepierre, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Statistical inference for the optimal approximating model

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ ²-distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.