Error estimation and adaptivity for pgd solutions

• Autores: Jonatha Oliveira de Matos Reis
• Directores de la Tesis: Pedro Díez Mejía (dir. tes.) , Jose Paulo Baptista Moitinho de Almeida (codir. tes.) , Sergio Zlotnik (codir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
• Idioma: español
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• Resumen

  • A significant part of the current research efforts in computational mechanics is focused on analyzing and handling large amounts of data, exploring high dimensional parametric spaces and providing answers to increasingly complex problems in short or real-time. This alludes to concepts (like digital twins) and technologies (like machine learning), methodologies to be considered in combination with classical computational models. Reduced Order Models (ROM) contribute to address these challenges by reducing the number of degrees of freedom of the models, suppressing redundancies in the description of the system to be modeled and simplifying the representation of the mathematical objects quantifying the physical magnitudes.

    Among these reduce order models, the Proper Generalized Decomposition (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected by the "curse of dimensionality", providing explicit expressions, computed a priori, of the parametric solution, making it is well suited to provide real-time responses. The PGD is a well-established reduced order method, but assessing the accuracy of its solutions is still a relevant challenge, particularly when seeking guaranteed bounds of the error of the solutions it provides.

    There are several approaches to analyze the errors of approximate solutions, but the only way that provides computable and guaranteed error bounds is by applying dual analysis. The idea behind dual analysis is to use a pair of complementary solutions (one that is compatible, and the other equilibrated) for a specified problem and to use the difference between these solutions, which bounds their errors. Dual analysis is also effective to drive mesh adaptivity refinement processes, as it provides information of the contribution of the elements to the error, either in a global or in a local framework.

    In this work we deal with finite element solutions for solid mechanics problems, computing compatible and equilibrated PGD solutions, using them in the context of dual analysis. The PGD approximations are obtained with an algebraic approach, leading to separable solutions that can be manipulated for an efficient computational implementation. We use these solutions to obtain global and local error bounds and use these bounds to drive an adaptivity process. The meshes obtained through these refinements provide solutions with errors significantly lower than those obtained using a uniform refinement.