Pierre-Eric Allier

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

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.

A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

Abstract We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems.

A New PGD Approximation: Computation and Verification

In this paper, a proper generalized decomposition (PGD) model reduction strategy is defined which is controlled and driven by means of the constitutive relation error (CRE). Because the main drawback of the PGD technique is the absence of a robust a posteriori error estimator to measure the quality of the approximate solution, such a strategy allows to use the power of the CRE to take into account all sources of error such as the discretization and modal truncation ones. The key point of that method is the construction of an admissible solution by post-processing the PGD one. To reuse the classical CRE technique, a first step consists of establishing a solution that respects the finite element equilibration. The main and only difficulty with the PGDapproximations is that they do not satisfy the finite element equilibrium. Then, to overcome this difficulty, a rather simple technique, is proposed, which associates the PGD-approximation and the data to a new approximation able to enter in the CREmachinery. Then the specific error indicators can be used in an adaptive strategy to construct a guaranteed solution up to a specific precision, and therefore to provide for reliable virtual charts for engineering design purposes.