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We present an apercu of our variational multiscale theory of LES turbulence. The theory is succinctly summarized in terms of a finite-dimensional coarse-scale equation governing the resolved scales that depend parametrically upon unresolved fine scales, which in turn are defined in terms of a functional of the coarse-scale residual ˵lifted″ to the dual of the fine-scale space, and the coarse-scale velocity field itself.We illustrate the performance of a numerical implementation of the theory with calculations of a turbulent channel flow at a friction-velocity Reynolds number of 395 and comparisons with the DNS data. [ABSTRACT FROM AUTHOR]

In isogeometric analysis (IGA for short) framework, computational domain is exactly described using the same representation as that employed in the CAD process. For a CAD object, we can construct various computational domain with same shape but with different parameterization. One basic requirement is that the resulting parameterization should have no self-intersections. In this paper, a linear and easy-to-check sufficient condition for injectivity of planar B-spline parameterization is proposed. By an example of 2D thermal conduction problem, we show that different parameterization of computational domain has different impact on the simulation result and efficiency in IGA. For problems with exact solutions, we propose a shape optimization method to obtain optimal parameterization of computational domain. The proposed injective condition is used to check the injectivity of initial parameterization constructed by discrete Coons method. Several examples and comparisons are presented to show the effectiveness of the proposed method. Compared with the initial parameterization during refinement, the optimal parameterization can achieve the same accuracy but with less degrees of freedom. [ABSTRACT FROM AUTHOR]

Isogeometric Analysis uses NURBS representations of the domain for performing numerical simulations. The first part of this paper presents a variational framework for generating NURBS parameterizations of swept volumes. The class of these volumes covers a number of interesting free-form shapes, such as blades of turbines and propellers, ship hulls or wings of airplanes. The second part of the paper reports the results of isogeometric analysis which were obtained with the help of the generated NURBS volume parameterizations. In particular we discuss the influence of the chosen parameterization and the incorporation of boundary conditions. [ABSTRACT FROM AUTHOR]

Chapter 10 of the book "Radon Transforms and the Rigidity of the Grassmannians," by Jacques Gasqui and Hubert Goldschmidt is presented. It demonstrates the non-rigidity and the geometry of the product of irreducible symmetric spaces. It examines the infinitesimal and Guillemin rigidity and products of symmetric spaces and conformally flat symmetric spaces. It also presents propositions which prove that Riemannian manifolds and symmetric spaces are locally isometric to two manifolds.