Krylov subspace recycling for evolving structures

Journal Article · 13 January 2022 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2021.114222· OSTI ID:1882856

Bolten, Matthias ^[1]; de Sturler, Eric ^[2]; Hahn, C. ^[1]; Parks, Michael L. ^[3]

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1. Univ. of Wuppertal (Germany)
2. Virginia Polytechnic Inst. and State Univ. (Virginia Tech), Blacksburg, VA (United States)
3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Krylov subspace recycling is a powerful tool when solving a long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these series appear naturally, as typically hundreds or thousands of optimization steps are needed with only small changes in the geometry. In this setting, however, applying Krylov subspace recycling can be a difficult task. As the geometry evolves, in general, so does the finite element mesh defined on or representing this geometry, including the numbers of nodes and elements and element connectivity. This is especially the case if re-meshing techniques are used. As a result, the number of algebraic degrees of freedom in the system changes, and in general the linear system matrices resulting from the finite element discretization change size from one optimization step to the next. Changes in the mesh connectivity also lead to structural changes in the matrices. In the case of re-meshing, even if the geometry changes only a little, the corresponding mesh might differ substantially from the previous one. Obviously, this prevents any straightforward mapping of the approximate invariant subspace of the linear system matrix (the focus of recycling in this work) from one optimization step to the next; similar problems arise for other selected subspaces. In this paper, we present an algorithm to map an approximate invariant subspace of the linear system matrix for the previous optimization step to an approximate invariant subspace of the linear system matrix for the current optimization step, for general meshes. This is achieved by exploiting the map from coefficient vectors to finite element functions on the mesh, combined with interpolation or approximation of functions on the finite element mesh. We demonstrate the effectiveness of our approach numerically with several proof of concept studies for a specific meshing technique.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: Federal Ministry of Research and Education of Germany; National Science Foundation (NSF); USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: NA0003525
• Other Award/Contract Number: 05M18PXA; 1720305
• OSTI ID: 1882856
• Report Number(s): SAND2021-13502J; 701783
• Journal Information: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 391; ISSN 0045-7825
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
Krylov subspace recycling
computations on changing geometries and structures
recycling CG
shape optimization