A New PGD Approximation: Computation and Verification

Abstract

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.