1,489 research outputs found
Model Reduction for Multiscale Lithium-Ion Battery Simulation
In this contribution we are concerned with efficient model reduction for
multiscale problems arising in lithium-ion battery modeling with spatially
resolved porous electrodes. We present new results on the application of the
reduced basis method to the resulting instationary 3D battery model that
involves strong non-linearities due to Buttler-Volmer kinetics. Empirical
operator interpolation is used to efficiently deal with this issue.
Furthermore, we present the localized reduced basis multiscale method for
parabolic problems applied to a thermal model of batteries with resolved porous
electrodes. Numerical experiments are given that demonstrate the reduction
capabilities of the presented approaches for these real world applications
eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation
Lattice networks with dissipative interactions are often employed to analyze
materials with discrete micro- or meso-structures, or for a description of
heterogeneous materials which can be modelled discretely. They are, however,
computationally prohibitive for engineering-scale applications. The
(variational) QuasiContinuum (QC) method is a concurrent multiscale approach
that reduces their computational cost by fully resolving the (dissipative)
lattice network in small regions of interest while coarsening elsewhere. When
applied to damageable lattices, moving crack tips can be captured by adaptive
mesh refinement schemes, whereas fully-resolved trails in crack wakes can be
removed by mesh coarsening. In order to address crack propagation efficiently
and accurately, we develop in this contribution the necessary generalizations
of the variational QC methodology. First, a suitable definition of crack paths
in discrete systems is introduced, which allows for their geometrical
representation in terms of the signed distance function. Second, special
function enrichments based on the partition of unity concept are adopted, in
order to capture kinematics in the wakes of crack tips. Third, a summation rule
that reflects the adopted enrichment functions with sufficient degree of
accuracy is developed. Finally, as our standpoint is variational, we discuss
implications of the mesh refinement and coarsening from an energy-consistency
point of view. All theoretical considerations are demonstrated using two
numerical examples for which the resulting reaction forces, energy evolutions,
and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after
review, paper title change
Randomized Local Model Order Reduction
In this paper we propose local approximation spaces for localized model order
reduction procedures such as domain decomposition and multiscale methods. Those
spaces are constructed from local solutions of the partial differential
equation (PDE) with random boundary conditions, yield an approximation that
converges provably at a nearly optimal rate, and can be generated at close to
optimal computational complexity. In many localized model order reduction
approaches like the generalized finite element method, static condensation
procedures, and the multiscale finite element method local approximation spaces
can be constructed by approximating the range of a suitably defined transfer
operator that acts on the space of local solutions of the PDE. Optimal local
approximation spaces that yield in general an exponentially convergent
approximation are given by the left singular vectors of this transfer operator
[I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However,
the direct calculation of these singular vectors is computationally very
expensive. In this paper, we propose an adaptive randomized algorithm based on
methods from randomized linear algebra [N. Halko et al. 2011], which constructs
a local reduced space approximating the range of the transfer operator and thus
the optimal local approximation spaces. The adaptive algorithm relies on a
probabilistic a posteriori error estimator for which we prove that it is both
efficient and reliable with high probability. Several numerical experiments
confirm the theoretical findings.Comment: 31 pages, 14 figures, 1 table, 1 algorith
Localized model reduction for parameterized problems
In this contribution we present a survey of concepts in localized model order
reduction methods for parameterized partial differential equations. The key
concept of localized model order reduction is to construct local reduced spaces
that have only support on part of the domain and compute a global approximation
by a suitable coupling of the local spaces. In detail, we show how optimal
local approximation spaces can be constructed and approximated by random
sampling. An overview of possible conforming and non-conforming couplings of
the local spaces is provided and corresponding localized a posteriori error
estimates are derived. We introduce concepts of local basis enrichment, which
includes a discussion of adaptivity. Implementational aspects of localized
model reduction methods are addressed. Finally, we illustrate the presented
concepts for multiscale, linear elasticity and fluid-flow problems, providing
several numerical experiments.
This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A.
Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order
Reduction. Walter De Gruyter GmbH, Berlin, 2019+
Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems
This is the accepted version of the following article: [Canales, D., Leygue, A., Chinesta, F., González, D., Cueto, E., Feulvarch, E., Bergheau, J. -M., and Huerta, A. (2016) Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems. Int. J. Numer. Meth. Engng, 108: 971–989. doi: 10.1002/nme.5240.], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5240/fullThis paper proposes a generalized finite element method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line and stored in memory in the form of a computational vademecum so that they can be used on-line with negligible cost. This renders a more efficient computational method than traditional finite element methods at performing simulations of processes. One key issue of the proposed method is the efficient computation of the parametric enrichments. These are computed and efficiently stored in memory by employing proper generalized decompositions. Although the presented method can be broadly applied, it is particularly well suited in manufacturing processes involving localized physics that depend on many parameters, such as welding. After introducing the vademecum-generalized finite element method formulation, we present some numerical examples related to the simulation of thermal models encountered in welding processes.Peer ReviewedPostprint (author's final draft
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