Error estimation and adaptivity for PGD based on complementary solutions applied to a simple 1D problem

Reduced order methods are powerful tools for the design and analysis of sophisticated systems, reducing computational costs and speeding up the development process. Among these reduced order methods, the Proper Generalized Decomposition is a well-established one, commonly used to deal with multi-dimensional problems that often suffer from the curse of dimensionality. Although the PGD method has been around for some time now, it still lacks mechanisms to assess the quality of the solutions obtained. This paper explores the dual error analysis in the scope of the PGD, using complementary solutions to compute error bounds and drive an adaptivity process, applied to a simple 1D problem. The energy of the error obtained from the dual analysis is used to determine the quality of the PGD approximations. We define a new adaptivity indicator based on the energy of the error and use it to drive parametric h- and p- adaptivity processes. The results are positive, with the indicator accurately capturing the parameter that will lead to lowest errors.

Error Estimation and Adaptivity for PGD based on Complementary Solutions

Reduced order methods are powerful tools for the design and analysis of sophisticated systems, reducing computational costs and speeding up the development process. Among these reduced order methods, the Proper Generalized Decomposition is a well-established one, commonly used to deal with multi-dimensional problems that often suffer from the curse of dimensionality. Although the PGD method has been around for some time now, it still lacks mechanisms to assess the quality of the solutions obtained. This paper explores the dual error analysis in the scope of the PGD, using complementary solutions to compute error bounds and drive an adaptivity process. The energy of the error obtained from the dual analysis is used to determine the quality of the PGD approximations. We define a new adaptivity indicator based on the energy of the error and use it to drive parametric h- and p- adaptivity processes. The results are positive, with the indicator accurately capturing the parameter that will lead to lowest errors.

Real time solutions of Thermo-Hydro Mechanical problems with application to the design of Engineered Barriers via Reduced Order Methods

<div><strong>Keywords:</strong> Thermo Hydro-Mechanical (THM), Model Order Reduction (MOR), Parametric solutions, Real time simulations.</div><div> </div><div>Radioactive waste is a by-product of nuclear power generation. It is hazardous to all forms of life and the environment. Its radioactive activity naturally decays over time, so waste has to be isolated and confined in appropriate disposal facilities for a sufficient period until it no longer poses a threat. Deep geological repositories constitute one of the most promising options for isolating this type of waste from human and environmental interactions. The analysis and prediction of the behaviour of such systems relies on coupled THM models [1]. The coupled nature of the problem is explained as follows [2]: i) a thermal part including the heat released by the wastes; ii) the mechanical behavior of the canister holding the wastes, the isolation system and the underground host rock; and, iii) the flow of natural water present in any underground porous media.</div><div><br>A coupled THM problem depends on space, time, and on material parameters (for instance, elastic modulus (E), heat conductivity (κ) and hydraulic conductivity (K)) and geometric parameters (for instance, the distance between canisters). We seek for families of solutions depending on these parameters. We would like to provide a real time numerical simulation of the THM problem for any value of the parameters within a range. Real time here, means a solution provided in a few seconds (instead of several hours). Such a solution can be used within an inversion problem, to obtain an best fitting value of the parameters based on some observations, or even in a control situation, where the prediction of the simulation is used to take some decision in the field.</div><div> </div><div>Reduced Order Methods (ROM)  are a family of numerical methods able to provide such a solutions. In this work we will present several parametric problems, and show how ROM  [3] can provide real time solution to (simple) THM problems.</div><div> </div><div> <p><strong>REFERENCES:</strong></p> <p>[1] Toprak, E.; Mokni, N.; Olivella, S.; Pintado, X.: Thermo-Hydro-Mechanical Modelling of Buffer. Synthesis Report. August 2013.</p> <p>[2] Selvadurai, A.P.S; Suvorov, A.P.: Thermo-Poroelasticity and Geomechanics.CAMBRIDGE UNIVERSITY PRESS, 2017.</p> <p>[3] Diez, P.; Zlotnik, S.; Garcia-Gonzalez, A.; Huerta, A.: Encapsulated PGD algebraic toolbox operating with high-dimensional data. Accepted In Archives of Computational Methods in Engineering, 2019.</p> </div>