An efficient quasi-optimal space-time PGD application to frictional contact mechanics

Anthony Giacoma,David Dureisseix,Anthony Gravouil

doi:10.1186/s40323-016-0067-7

Abstract

The proper generalized decomposition (PGD) aims at finding the solution of a generic problems into a low rank approximation. On the contrary to the singular value decomposition (SVD), such a low rank approximation is generally not the optimal one leading to memory issues and loss of computational efficiency. Nonetheless, the computational cost of the SVD is generally prohibitive to be performed. In this paper, authors suggest an algorithm to address this issue. First, the algorithm is described and studied in details. It consists in a cheap iterative method compressing a low rank expansion. It will be shown that given a low rank approximation, the SVD of a provided low rank approximation can be reached at convergence. Behavior of the method is exhibited on a numerical application. Second, the algorithm is embedded into a general space-time PGD solver to compress the iterated separated form for the solution. An application to a quasi-static frictional contact problem is illustrated. Then, efficiency of such a compressing method will be demonstrated.

Highlights

Computational mechanics tackles nowadays large models involving huge amount of data to provide fine description of physics or accurate forecasts
We propose to illustrate the previously described algorithm into a space-time proper generalized decomposition (PGD) solver aiming at solving a frictional contact solid mechanic quasi-static problem
The LArge Time INcrement (LATIN)-PGD computes painlessly dominant trends and iterated vector are very close to singular value decomposition (SVD) optimal vectors of the solution

Summary

Background

Computational mechanics tackles nowadays large models involving huge amount of data to provide fine description of physics or accurate forecasts. Several and various numerical methods have to be taken into account in order to perform efficiently these large scale simulations (both accurate and computationally cheap) To address this issue, both computational hardware and algorithms have to progress. A large amount of computational time can be spared and accurate and well representative solution can be captured These methods rely strongly on basis design for approximated solution which has to span the dominant trends and perhaps weaker ones up to a desired level of accuracy. Such an approach ensures to have at each iteration a basis which spans the whole considered space to detriment of its optimality Such an iterated basis could be sufficient to perform reliable computation or data analysis. The efficiency of quasi-optimal approaches will be exemplified

An iterative process to compute the SVD

Definition of the compression function F

Algorithm properties

Generalization to higher rank expansions

Rank adaptation and downsizing

Conclusion