Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures

Abstract

The paper deals with the constitutive relation error (CRE) concept which has been widely used over the last 40 years for verification and validation of computational mechanics models. It more specifically focuses on the beneficial use of model reduction based on proper generalized decomposition (PGD) into this CRE concept. Indeed, it is shown that a PGD formulation can facilitate the construction of so-called admissible fields which is a technical key-point of CRE. Numerical illustrations, addressing both model verification and model updating, are presented to assess the performances of the proposed approach.

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

“…These approximated displacements v h correspond to the Lagrange multipliers of constraints (14) and (15). The bases for the displacements on all the edges are assigned as columns of V and the weights defining their combination are represented inv.…”

“…We can now combine Equation (12) with the constrains (14). Substituting the approximations (17) and (16) into (12) and (16) into (14), and minimizing the resulting energy leads to the following equations:…”

“…12 There are recent applications of this technique, namely the work by Chamoin and Thai, exploring isogeometric analysis and PGD solutions to deal with geometric parameters and applying the constitute relation error as an error estimator and indicator for mesh refinement. [13][14][15][16] Another approach using the dual error analysis concept was performed by Moitinho, using the extended formulation from Prager to general problems proposed by Debongnie and using an independent PGD solution, based on hybrid equilibrium finite elements, to define the equilibrated stress field. [17][18][19][20] When analyzing the errors of a solutions, it is often desirable to have information of specific quantities of interest (QoI).…”

Fast model updating coupling Bayesian inference and PGD model reduction

“…The PGD method was introduced in [29] as "radial approximation" to solve nonlinear problems in structural mechanics. Since [19], this method was used in many fields: model verification and validation [18], virtual charts for the engineering [20] [21] etc. PGD is also used for identification problems in a deterministic framework [25] [10] [3] and the great possible number of parameters types to use in those models [8] seems to be well suited for Bayesian inference.…”

Towards simplified and optimized a posteriori error estimation using PGD reduced models

“…First, starting from the classical hybrid‐flux approach, we introduce a reduced model at the element level issued from a parameterization of local problems. This approach, later referred as EET‐PGD, was primarily introduced in Chamoin et al for a simple case (it is generalized here). It leads to a precomputation of local equilibrium from given data (tractions, element geometry, and material), in an offline step, using PGD.…”