Abstract
This work investigates matrix-free algorithms for problems in quasi-static finite-strain hyperelasticity. Iterative solvers with matrix-free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix-free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. The performance of the matrix-free operator and the geometric multigrid preconditioner is compared to the matrix-based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix-free methods for finite-strain solid mechanics are very promising, outperforming linear matrix-based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.
| Original language | English |
|---|---|
| Pages (from-to) | 2874-2895 |
| Number of pages | 22 |
| Journal | International Journal for Numerical Methods in Engineering |
| Volume | 121 |
| Issue number | 13 |
| DOIs | |
| State | Published - Jul 15 2020 |
Funding
D. Davydov acknowledges the financial support of the German Research Foundation (DFG), grant DA 1664/2‐1. D. Davydov, D. Arndt, and J‐P. Pelteret are grateful to Jed Brown (CU Boulder) and Veselin Dobrev (Lawrence Livermore National Laboratory) for fruitful discussions on matrix‐free operator evaluation approaches. D. Arndt and M. Kronbichler were supported by the German Research Foundation (DFG) under the project “High‐order discontinuous Galerkin for the exa‐scale” (ExaDG), grant 279336170 within the priority program “Software for Exascale Computing” (SPPEXA). P. Steinmann acknowledges the support of the Cluster of Excellence Engineering of Advanced Materials (EAM) which made this collaboration possible, as well as funding by the EPSRC Strategic Support Package “Engineering of Active Materials by Multiscale/Multiphysics Computational Mechanics” (EP/R008531/1). This manuscript has been authored by UT‐Battelle, LLC under Contract No. DE‐AC05‐00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non‐exclusive, paid‐up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan ( http://energy.gov/downloads/doe‐public‐access‐plan ). information Cluster of Excellence Engineering of Advanced Materials (EAM), Deutsche Forschungsgemeinschaft, 279336170; DA 1664/2-1; Engineering and Physical Sciences Research Council, EP/R008531/1D.?Davydov acknowledges the financial support of the German Research Foundation (DFG), grant DA 1664/2-1. D.?Davydov, D.?Arndt, and J-P.?Pelteret are grateful to Jed Brown (CU Boulder) and Veselin Dobrev (Lawrence Livermore National Laboratory) for fruitful discussions on matrix-free operator evaluation approaches. D.?Arndt and M.?Kronbichler were supported by the German Research Foundation (DFG) under the project ?High-order discontinuous Galerkin for the exa-scale? (ExaDG), grant 279336170 within the priority program ?Software for Exascale Computing? (SPPEXA). P.?Steinmann acknowledges the support of the Cluster of Excellence Engineering of Advanced Materials (EAM) which made this collaboration possible, as well as funding by the EPSRC Strategic Support Package ?Engineering of Active Materials by Multiscale/Multiphysics Computational Mechanics? (EP/R008531/1). This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
| Funders | Funder number |
|---|---|
| DOE Public Access Plan | |
| ExaDG | |
| United States Government | |
| Veselin Dobrev | |
| U.S. Department of Energy | |
| Engineering and Physical Sciences Research Council | DE-AC05-00OR22725, EP/R008531/1 |
| Deutsche Forschungsgemeinschaft | DA 1664/2‐1, 279336170 |
Keywords
- adaptive finite-element method
- finite-strain
- geometric multigrid
- hyperelasticity
- matrix-free