189 research outputs found
Some remarks about the positivity of random variables on a Gaussian probability space
Let be an abstract Wiener space and be a probability density
of class LlogL. Using the measure transportation of Monge-Kantorovitch, we
prove that the kernel of the projection of L on the second Wiener chaos defines
an (Hilbert-Schmidt) operator which is lower bounded by another Hilbert-Schmidt
operator.Comment: 6 page
Computational Crystal Plasticity: From Single Crystal to Homogenized Polycrystals
Crystal plasticity models for single crystals at large deformation are shown. An extension to the computation of polycrystals is also proposed. The scale transition rule is numerically identified on polycrystal computations, and is valid for any type of loading. All these models are implemented in a finite element code, which has a sequential and a parallel version. Parallel processing makes CPU time reasonable, even for 3D meshes involving a large number of internal variables (more than 1000) at each Gauss point.Together with a presentation of the numerical tools, the paper shows several applications, a study of the crack tip strain fields in single crystals, of zinc coating on a steel substrate, specimen computation involving a large number of grains in each Gauss point. Finally, polycrystalline aggregates are generated, and numerically tested. The effect of grain boundary damage, opening and sliding is investigated
The Monge problem in Wiener Space
We address the Monge problem in the abstract Wiener space and we give an
existence result provided both marginal measures are absolutely continuous with
respect to the infinite dimensional Gaussian measure {\gamma}
Bridging Proper Orthogonal Decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems
This article describes a bridge between POD-based model order reduction
techniques and the classical Newton/Krylov solvers. This bridge is used to
derive an efficient algorithm to correct, "on-the-fly", the reduced order
modelling of highly nonlinear problems undergoing strong topological changes.
Damage initiation problems are addressed and tackle via a corrected
hyperreduction method. It is shown that the relevancy of reduced order model
can be significantly improved with reasonable additional costs when using this
algorithm, even when strong topological changes are involved
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
This paper is devoted to a deeper understanding of the heat flow and to the
refinement of calculus tools on metric measure spaces (X,d,m). Our main results
are:
- A general study of the relations between the Hopf-Lax semigroup and
Hamilton-Jacobi equation in metric spaces (X,d).
- The equivalence of the heat flow in L^2(X,m) generated by a suitable
Dirichlet energy and the Wasserstein gradient flow of the relative entropy
functional in the space of probability measures P(X).
- The proof of density in energy of Lipschitz functions in the Sobolev space
W^{1,2}(X,d,m).
- A fine and very general analysis of the differentiability properties of a
large class of Kantorovich potentials, in connection with the optimal transport
problem.
Our results apply in particular to spaces satisfying Ricci curvature bounds
in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the
doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4,
Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop.
4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6
simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients,
still equivalent to all other ones, has been propose
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