A hyper-reduction method using adaptivity to cut the assembly costs of reduced order models

Computer Methods in Applied Mechanics and Engineering, vol. 380, pp. 113723

Abstract

At every iteration or timestep of the online phase of some reduced-order modelling schemes for non-linear or time-dependent systems, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The projected small linear systems are cheap to solve, but assembly and projection become the dominant computational cost. In this paper we introduce a new hyper-reduction strategy called reduced assembly (RA) that drastically cuts these costs. RA consists of a triangulation adaptation algorithm that uses a local error indicator to construct a reduced assembly triangulation specially suited to the reduced order basis. Crucially, this reduced assembly triangulation has fewer cells than the original one, resulting in lower assembly and projection costs. We demonstrate the efficacy of RA on a Galerkin-POD type reduced order model (RAPOD). We show performance increases up to five times over the baseline Galerkin-POD method on a non-linear reaction–diffusion problem solved with a semi-implicit time-stepping scheme and up to seven times for a 3D hyperelasticity problem solved with a continuation Newton–Raphson algorithm. The examples are implemented in the DOLFIN finite element solver using PETSc and SLEPc for linear algebra. Full code and data files to produce the results in this paper are provided as supplementary material.

Authors 5

1. Jack Hale

   University of Luxembourg

   Affiliation as printed

   Institute of Computational Engineering, University of Luxembourg, Luxembourg

2. Elisa Schenone

   University of Luxembourg

   Affiliation as printed

   Institute of Computational Engineering, University of Luxembourg, Luxembourg

3. Davide Baroli Aachen Aachen Institute for Advanced Study in Computational Engineering Science (AICES)

   University of Luxembourg · Aachen Institute for Advanced Study in Computational Engineering Science · RWTH Aachen University

   Affiliation as printed

   Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University, Germany

   Institute of Computational Engineering, University of Luxembourg, Luxembourg

4. Lars Beex

   University of Luxembourg

   Affiliation as printed

   Institute of Computational Engineering, University of Luxembourg, Luxembourg

5. Stéphane Bordas corresponding

   University of Luxembourg · Cardiff University

   Affiliation as printed

   Institute of Computational Engineering, University of Luxembourg, Luxembourg

   School of Engineering, Cardiff University, United Kingdom