Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

Author (s): Reis, J.; Moitinho de Almeida, J.P.; Díez, P. and Zlotnik, S.
Journal: International Journal for Numerical Methods in Engineering

Volume: 122
Pages: 752 – 776
Date: 2021

Abstract:
When designing a structure or engineering a component, it is crucial to use methods that provide fast and reliable
solutions, so that a large number of design options can be assessed. In this context, the Proper Generalized Decomposition
can be a powerful tool, as it provides solutions to parametric problems, without being aected by the “curse of
dimensionality”.
Assessing the accuracy of the solutions obtained with the PGD is still a relevant challenge, particularly when seeking
quantities of interest with guaranteed bounds.
In this work, we compute compatible and equilibrated PGD solutions and use them in a dual analysis to obtain
quantities of interest and their bounds, which are guaranteed.
We also use these complementary solutions to compute an error indicator, which is used to drive a mesh adaptivity
process, oriented for a quantity of interest. The corresponding solutions have errors that are much lower than those
obtained using a uniform renement or an indicator based on the global error, as the proposed approach focuses on
minimizing the error in the quantity of interest.

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Bibtex:

@article{2021-IJNME-RMDZ,
        Author = {Reis, Jonatha
and Moitinho de Almeida, Jose Paulo
and D{\'i}ez, Pedro
and Zlotnik, Sergio},
        Title = {Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest},
        Fjournal = {International Journal for Numerical Methods in Engineering},
        Volume = {122},
        Number = {3},
        Pages = {752-776 },
        Year = {2021},
         Doi = {10.1002/nme.6559},
        }