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Algebraic Multigrid Based Preconditioning Approaches for Generalized Continuum Models and Indirect Displacement Control Techniques.

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Title: Algebraic Multigrid Based Preconditioning Approaches for Generalized Continuum Models and Indirect Displacement Control Techniques.
Authors: Alkmim, Nasser1 (AUTHOR) nasser.alkmim@uibk.ac.at, Gamnitzer, Peter1 (AUTHOR), Dummer, Alexander1 (AUTHOR), Neuner, Matthias2 (AUTHOR), Hofstetter, Günter1 (AUTHOR)
Source: International Journal for Numerical Methods in Engineering. 3/30/2026, Vol. 127 Issue 6, p1-32. 32p.
Subjects: Algebraic multigrid methods, Continuum mechanics, Numerical analysis, Brittle materials
Abstract: The contribution deals with algebraic multigrid (AMG) based preconditioning methods for the iterative solution of a coupled linear system of equations arising in numerical simulations of failure of quasi‐brittle materials using generalized continuum approaches. In particular, the focus is on the solution of large and sparse linear systems originating from a gradient‐enhanced micropolar formulation with coupled fields of displacements, microrotations, and non‐local damage. Moreover, due to the possible presence of snap‐back behavior in quasi‐brittle materials, indirect displacement control techniques are discussed in the context of iterative linear solvers. The respective linear systems exhibit a distinct block structure representing the coupled fields of unknowns, which requires specialized preconditioners to iteratively solve the fully coupled linear system. Firstly, the present paper describes and investigates monolithic multigrid strategies for such problems which treat the block structure within the AMG hierarchy. An evaluation of the performance of the monolithic multigrid strategy for displacement controlled problems is carried out. This is done by a comparison to the performance of a previously published AMG based preconditioning strategy that applies the AMG to each field separately in a block preconditioning fashion. The results obtained for 2D plane strain and 3D triaxial compression indicate that this monolithic multigrid strategy performs similarly to the reference approach in terms of iteration counts in the majority of the investigated test cases. However, for distinct choices of material parameters, the respective strategy is shown to outperform the reference approach prior to localization and damage initiation. Secondly, for problems relying on indirect displacement control, a novel monolithic solution scheme is proposed that extends the existing block preconditioner to accommodate the additional constraint equation. We present a block preconditioner for the augmented system, for which we show in a 2D simulation of borehole failure that it significantly enhances computational efficiency by combining indirect displacement control and iterative linear solution techniques. [ABSTRACT FROM AUTHOR]
Copyright of International Journal for Numerical Methods in Engineering is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Algebraic Multigrid Based Preconditioning Approaches for Generalized Continuum Models and Indirect Displacement Control Techniques.
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+for+Numerical+Methods+in+Engineering%22">International Journal for Numerical Methods in Engineering</searchLink>. 3/30/2026, Vol. 127 Issue 6, p1-32. 32p.
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  Data: <searchLink fieldCode="DE" term="%22Algebraic+multigrid+methods%22">Algebraic multigrid methods</searchLink><br /><searchLink fieldCode="DE" term="%22Continuum+mechanics%22">Continuum mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Brittle+materials%22">Brittle materials</searchLink>
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  Data: The contribution deals with algebraic multigrid (AMG) based preconditioning methods for the iterative solution of a coupled linear system of equations arising in numerical simulations of failure of quasi‐brittle materials using generalized continuum approaches. In particular, the focus is on the solution of large and sparse linear systems originating from a gradient‐enhanced micropolar formulation with coupled fields of displacements, microrotations, and non‐local damage. Moreover, due to the possible presence of snap‐back behavior in quasi‐brittle materials, indirect displacement control techniques are discussed in the context of iterative linear solvers. The respective linear systems exhibit a distinct block structure representing the coupled fields of unknowns, which requires specialized preconditioners to iteratively solve the fully coupled linear system. Firstly, the present paper describes and investigates monolithic multigrid strategies for such problems which treat the block structure within the AMG hierarchy. An evaluation of the performance of the monolithic multigrid strategy for displacement controlled problems is carried out. This is done by a comparison to the performance of a previously published AMG based preconditioning strategy that applies the AMG to each field separately in a block preconditioning fashion. The results obtained for 2D plane strain and 3D triaxial compression indicate that this monolithic multigrid strategy performs similarly to the reference approach in terms of iteration counts in the majority of the investigated test cases. However, for distinct choices of material parameters, the respective strategy is shown to outperform the reference approach prior to localization and damage initiation. Secondly, for problems relying on indirect displacement control, a novel monolithic solution scheme is proposed that extends the existing block preconditioner to accommodate the additional constraint equation. We present a block preconditioner for the augmented system, for which we show in a 2D simulation of borehole failure that it significantly enhances computational efficiency by combining indirect displacement control and iterative linear solution techniques. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>Copyright of International Journal for Numerical Methods in Engineering is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1002/nme.70309
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        Text: English
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      – SubjectFull: Continuum mechanics
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      – TitleFull: Algebraic Multigrid Based Preconditioning Approaches for Generalized Continuum Models and Indirect Displacement Control Techniques.
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              Text: 3/30/2026
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