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Schwarz Inequality (schwarz + inequality)

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Engineering50%

Selected Abstracts

Convergence analysis and validation of sequential limit analysis of plane-strain problems of the von Mises model with non-linear isotropic hardening

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005
S.-Y. Leu
Abstract The paper presents sequential limit analysis of plane-strain problems of the von Mises model with non-linear isotropic hardening by using a general algorithm. The general algorithm is a combined smoothing and successive approximation (CSSA) method. In the paper, emphasis is placed on its convergence analysis and validation applied to sequential limit analysis involving materials with isotropic hardening. By sequential limit analysis, the paper treats deforming problems as a sequence of limit analysis problems stated in the upper bound formulation. Especially, the CSSA algorithm was proved to be unconditionally convergent by utilizing the Cauchy,Schwarz inequality. Finally, rigorous validation was conducted by numerical and analytical studies of a thick-walled cylinder under pressure. It is found that the computed limit loads are rigorous upper bounds and agree very well with the analytical solutions. Copyright © 2005 John Wiley & Sons, Ltd. [source]

On the multilevel preconditioning of Crouzeix,Raviart elliptic problems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2008
J. Kraus
Abstract We consider robust hierarchical splittings of finite element spaces related to non-conforming discretizations using Crouzeix,Raviart type elements. As is well known, this is the key to the construction of efficient two- and multilevel preconditioners. The main contribution of this paper is a theoretical and an experimental comparison of three such splittings. Our starting point is the standard method based on differences and aggregates (DA) as introduced in Blaheta et al. (Numer. Linear Algebra Appl. 2004; 11:309,326). On this basis we propose a more general (GDA) splitting, which can be viewed as the solution of a constraint optimization problem (based on certain symmetry assumptions). We further consider the locally optimal (ODA) splitting, which is shown to be equivalent to the first reduce (FR) method from Blaheta et al. (Numer. Linear Algebra Appl. 2004; 11:309,326). This means that both, the ODA and the FR splitting, generate the same subspaces, and thus the local constant in the strengthened Cauchy,Bunyakowski,Schwarz inequality is minimal for the FR (respectively ODA) splitting. Moreover, since the DA splitting corresponds to a particular choice in the parameter space of the GDA splitting, which itself is an element in the set of all splittings for which the ODA (or equivalently FR) splitting yields the optimum, we conclude that the chain of inequalities ,,,,,,3/4 holds independently of mesh and/or coefficient anisotropy. Apart from the theoretical considerations, the presented numerical results provide a basis for a comparison of these three approaches from a practical point of view. Copyright © 2007 John Wiley & Sons, Ltd. [source]

A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2008
J. K. Kraus
Abstract We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant , in the strengthened Cauchy,Bunyakowski,Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright © 2007 John Wiley & Sons, Ltd. [source]

An eddy viscosity model with near-wall modifications

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2005
M. M. Rahman
Abstract An extended version of the isotropic k,, model is proposed that accounts for the distinct effects of low-Reynolds number (LRN) and wall proximity. It incorporates a near-wall correction term to amplify the level of dissipation in nonequilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and the Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Comparisons indicate that the present model is a significant improvement over the standard eddy viscosity formulation. Copyright © 2005 John Wiley & Sons, Ltd. [source]