349 research outputs found
An introduction to Lie group integrators -- basics, new developments and applications
We give a short and elementary introduction to Lie group methods. A selection
of applications of Lie group integrators are discussed. Finally, a family of
symplectic integrators on cotangent bundles of Lie groups is presented and the
notion of discrete gradient methods is generalised to Lie groups
Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program
Computer programs may go wrong due to exceptional behaviors, out-of-bound
array accesses, or simply coding errors. Thus, they cannot be blindly trusted.
Scientific computing programs make no exception in that respect, and even bring
specific accuracy issues due to their massive use of floating-point
computations. Yet, it is uncommon to guarantee their correctness. Indeed, we
had to extend existing methods and tools for proving the correct behavior of
programs to verify an existing numerical analysis program. This C program
implements the second-order centered finite difference explicit scheme for
solving the 1D wave equation. In fact, we have gone much further as we have
mechanically verified the convergence of the numerical scheme in order to get a
complete formal proof covering all aspects from partial differential equations
to actual numerical results. To the best of our knowledge, this is the first
time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with
arXiv:1112.179
Clustering based Multiple Anchors High-Dimensional Model Representation
In this work, a cut high-dimensional model representation (cut-HDMR)
expansion based on multiple anchors is constructed via the clustering method.
Specifically, a set of random input realizations is drawn from the parameter
space and grouped by the centroidal Voronoi tessellation (CVT) method. Then for
each cluster, the centroid is set as the reference, thereby the corresponding
zeroth-order term can be determined directly. While for non-zero order terms of
each cut-HDMR, a set of discrete points is selected for each input component,
and the Lagrange interpolation method is applied. For a new input, the cut-HDMR
corresponding to the nearest centroid is used to compute its response.
Numerical experiments with high-dimensional integral and elliptic stochastic
partial differential equation as backgrounds show that the CVT based multiple
anchors cut-HDMR can alleviate the negative impact of a single inappropriate
anchor point, and has higher accuracy than the average of several expansions
Sparse-grids uncertainty quantification of part-scale additive manufacturing processes
The present paper aims at applying uncertainty quantification methodologies
to process simulations of powder bed fusion of metal. In particular, for a
part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study
the uncertainties of three process parameters, namely the activation
temperature, the powder convection coefficient and the gas convection
coefficient. First, we perform a variance-based global sensitivity analysis to
study how each uncertain parameter contributes to the variability of the beam
displacements. The results allow us to conclude that the gas convection
coefficient has little impact and can therefore be fixed to a constant value
for subsequent studies. Then, we conduct an inverse uncertainty quantification
analysis, based on a Bayesian approach on synthetic displacements data, to
quantify the uncertainties of the two remaining parameters, namely the
activation temperature and the powder convection coefficient. Finally, we use
the results of the inverse uncertainty quantification analysis to perform a
data-informed forward uncertainty quantification analysis of the residual
strains. Crucially, we make use of surrogate models based on sparse grids to
keep to a minimum the computational burden of every step of the uncertainty
quantification analysis. The proposed uncertainty quantification workflow
allows us to substantially ease the typical trial-and-error approach used to
calibrate power bed fusion part-scale models, and to greatly reduce
uncertainties on the numerical prediction of the residual strains. In
particular, we demonstrate the possibility of using displacement measurements
to obtain a data-informed probability density function of the residual strains,
a quantity much more complex to measure than displacements
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