Error estimation for proper generalized decomposition solutions: A dual approach

Numerical Meth Engineering

Dı́ez

et al. 2020

Self Cite

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 (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected 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 refinement or an indicator based on the global error, as the proposed approach focuses on minimizing the error in the quantity of interest.

Section: Resultsmentioning

confidence: 99%

“…A more detailed explanation on the stopping criteria used can be found in our previous work on global adaptivity for PGD solutions. 29 The parametric domain, unless otherwise stated, has n h = 50 points. The tolerance for the PGD model, the fixed point iteration and the adaptivity process are all set to 10 −4 .…”

Section: Resultsmentioning

confidence: 99%

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

Numerical Meth Engineering

Dı́ez

et al. 2020

Self Cite

“…Additional work is being done to extend the results obtained to a 2D and 3D framework [13]. We can also extend the results to obtain quantities of interest and bounds of their errors from for the PGD approximations [14], giving a direct physical meaning to the solutions obtained.…”

Section: Resultsmentioning

confidence: 99%

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

Adv. Model. and Simul. in Eng. Sci.

Dı́ez

et al. 2020

Self Cite

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.

“…This technique was developed specifically for the application of the hybrid equilibrium formulation, 4 in the context of determining bounds of the error of the solutions obtained from reduced order models 10 . The characteristic of hybrid equilibrium elements are well documented, and a comprehensive study of the convergence of these elements applied to 3D elasticity problems can be found in Reference 11.…”

Section: Introductionmentioning

confidence: 99%

An efficient methodology for stress‐based finite element approximations in two‐dimensional elasticity

Numerical Meth Engineering

2020

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Summary We present an efficient approach to compute the matrices used in finite element formulations where self‐equilibrated (zero divergence) approximations of the stress field are used. The fundamental aspect of this approach is that it is applicable to polynomial approximations of high degree (it was tested up to degree 10), providing closed‐form expressions for the elementary matrices involved. The components of the stress field are defined as a function of the coordinates in the master element, facilitating the postprocessing of the finite element results and allowing for the consideration of geometric parameters in reduced order models.