5,591 research outputs found
Substructured formulations of nonlinear structure problems - influence of the interface condition
We investigate the use of non-overlapping domain decomposition (DD) methods
for nonlinear structure problems. The classic techniques would combine a global
Newton solver with a linear DD solver for the tangent systems. We propose a
framework where we can swap Newton and DD, so that we solve independent
nonlinear problems for each substructure and linear condensed interface
problems. The objective is to decrease the number of communications between
subdomains and to improve parallelism. Depending on the interface condition, we
derive several formulations which are not equivalent, contrarily to the linear
case. Primal, dual and mixed variants are described and assessed on a simple
plasticity problem.Comment: in International Journal for Numerical Methods in Engineering, Wiley,
201
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
The cost of continuity: performance of iterative solvers on isogeometric finite elements
In this paper we study how the use of a more continuous set of basis
functions affects the cost of solving systems of linear equations resulting
from a discretized Galerkin weak form. Specifically, we compare performance of
linear solvers when discretizing using B-splines, which span traditional
finite element spaces, and B-splines, which represent maximum
continuity. We provide theoretical estimates for the increase in cost of the
matrix-vector product as well as for the construction and application of
black-box preconditioners. We accompany these estimates with numerical results
and study their sensitivity to various grid parameters such as element size
and polynomial order of approximation . Finally, we present timing results
for a range of preconditioning options for the Laplace problem. We conclude
that the matrix-vector product operation is at most \slfrac{33p^2}{8} times
more expensive for the more continuous space, although for moderately low ,
this number is significantly reduced. Moreover, if static condensation is not
employed, this number further reduces to at most a value of 8, even for high
. Preconditioning options can be up to times more expensive to setup,
although this difference significantly decreases for some popular
preconditioners such as Incomplete LU factorization
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