635 research outputs found
Strict lower bounds with separation of sources of error in non-overlapping domain decomposition methods
This article deals with the computation of guaranteed lower bounds of the
error in the framework of finite element (FE) and domain decomposition (DD)
methods. In addition to a fully parallel computation, the proposed lower bounds
separate the algebraic error (due to the use of a DD iterative solver) from the
discretization error (due to the FE), which enables the steering of the
iterative solver by the discretization error. These lower bounds are also used
to improve the goal-oriented error estimation in a substructured context.
Assessments on 2D static linear mechanic problems illustrate the relevance of
the separation of sources of error and the lower bounds' independence from the
substructuring. We also steer the iterative solver by an objective of precision
on a quantity of interest. This strategy consists in a sequence of solvings and
takes advantage of adaptive remeshing and recycling of search directions.Comment: International Journal for Numerical Methods in Engineering, Wiley,
201
Improved recovery of admissible stress in domain decomposition methods - application to heterogeneous structures and new error bounds for FETI-DP
This paper investigates the question of the building of admissible stress
field in a substructured context. More precisely we analyze the special role
played by multiple points. This study leads to (1) an improved recovery of the
stress field, (2) an opportunity to minimize the estimator in the case of
heterogeneous structures (in the parallel and sequential case), (3) a procedure
to build admissible fields for FETI-DP and BDDC methods leading to an error
bound which separates the contributions of the solver and of the
discretization
Isogeometric Analysis and Harmonic Stator-Rotor Coupling for Simulating Electric Machines
This work proposes Isogeometric Analysis as an alternative to classical
finite elements for simulating electric machines. Through the spline-based
Isogeometric discretization it is possible to parametrize the circular arcs
exactly, thereby avoiding any geometrical error in the representation of the
air gap where a high accuracy is mandatory. To increase the generality of the
method, and to allow rotation, the rotor and the stator computational domains
are constructed independently as multipatch entities. The two subdomains are
then coupled using harmonic basis functions at the interface which gives rise
to a saddle-point problem. The properties of Isogeometric Analysis combined
with harmonic stator-rotor coupling are presented. The results and performance
of the new approach are compared to the ones for a classical finite element
method using a permanent magnet synchronous machine as an example
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
Domain decomposition preconditioners of Neumann-Neumann type for hpâapproximations on boundary layer meshes in three dimensions
We develop and analyse Neumann-Neumann methods for hp finiteâelement approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. These are meshes that are highly anisotropic where the aspect ratio typically grows exponentially with the polynomial degree. The condition number of our preconditioners is shown to be independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. In addition, it only grows polylogarithmically with the polynomial degree, as in the case of p approximations on shapeâregular meshes. This work generalizes our previous one on twoâdimensional problems in Toselli & Vasseur (2003a, submitted to Numerische Mathematik, 2003c to appear in Comput. Methods Appl. Mech. Engng.) and the estimates derived here can be employed to prove condition number bounds for certain types of FETI method
- âŠ