Effectivity and limits of PGD computational techniques

Mechanics, like other domains, continues to supply numerous engineering problems that, despite the impressive progress of computational simulation techniques, remain intractable today. RB, POD and PGD model reduction methods are leading to a new generation of high-performance computational tools which provide solutions to engineering problems which are inaccessible to standard codes based on classical and well-established numerical techniques. This is a true breakthrough with a potential gain of several orders of magnitude. The approach we are considering here is the Proper Generalized Decomposition (PGD) which can be seen as an extension of POD. The main idea consists in calculating the shape functions and the solution itself simultaneously offline using an iterative procedure. A priori, these shape functions are arbitrary and must satisfy only a variable separation hypothesis, this hypothesis being also at the center of POD and RB reduction methods. First, the aim of the lecture is to examine the validity of the variable separation hypothesis, which is central in the PGD. For that, we use a benchmark proposed by S. Idelshon which is a unidimensional transient thermal problem with a moving load, the parameter being the velocity of the load. Non-separated shape functions are also studied. The second part of the lecture deals with the different techniques for the computation of PGD approximation. They include a new one based on the “PGD-error indicator” introduced in [2,3]. The reference to quantify the effectivity is the H1-SVD of the “exact” solution. Some error estimators are also computed.